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Chuang, J. & Lazarev, A. (2011). L-infinity maps and twistings. Homology, Homotopy and
Applications, 13(2), pp. 175-195.
City Research Online
Original citation: Chuang, J. & Lazarev, A. (2011). L-infinity maps and twistings. Homology,
Homotopy and Applications, 13(2), pp. 175-195.
Permanent City Research Online URL: http://openaccess.city.ac.uk/768/
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L-INFINITY MAPS AND TWISTINGS
J. CHUANG AND A. LAZAREV
Abstract. We give a construction of an L-infinity map from any L-infinity algebra into its
truncated Chevalley-Eilenberg complex as well as its cyclic and A-infinity analogues. This map
fits with the inclusion into the full Chevalley-Eilenberg complex (or its respective analogues) to
form a homotopy fiber sequence of L-infinity algebras. Applications to deformation theory and
graph homology are given. We employ the machinery of Maurer-Cartan functors in L-infinity
and A-infinity algebras and associated twistings which should be of independent interest.
1. introduction
Let V be a Lie algebra and f : V → Hom(V, V ) be the map that associates to v ∈ V
the endomorphism f (v) := ad(v) : x 7→ [v, x] for x ∈ V . Then f is clearly a Lie algebra
homomorphism V → Der(V, V ) from V into the Lie algebra of derivations of V . If instead
V is an associative algebra, f is still a Lie algebra map from the commutator Lie algebra of
V into the Lie algebra of derivations of V . Furthermore, a variant of this construction works
when V has an invariant scalar product h, i, so that h[a, b], ci = ha, [b, c]i in the Lie case and
hab, ci = ha, bci in the associative case; here a, b, c ∈ V . In this case the image of f lands in the
subspace of Der(V, V ) consisting of skew self-adjoint endomorphisms of V .
One of the purposes of this paper is to generalize this simple and well-known construction
to L∞ - and A∞ -algebras. This generalization will be applied to characteristic classes of A∞ algebras in a future paper, but we believe that it is of independent interest. A homotopyinvariant analogue of Der(V, V ) for an L∞ -algebra V is a truncated Chevalley-Eilenberg complex
CE(V, V ) which is part of the following homotopy fiber sequence of complexes:
(1.1)
f
V
/ CE(V, V )
g
/ CE(V, V ),
where CE(V, V ) is the usual (untruncated) Chevalley-Eilenberg complex and f is a certain
generalization of the map f = ad defined above. Similarly, in the A∞ algebra context, a
homotopy-invariant analogue of Der(V, V ) is a truncated Hochschild complex Hoch(V, V ) which
is part of the following homotopy fiber sequence of complexes:
(1.2)
f
V
/ Hoch(V, V )
g
/ Hoch(V, V ),
where Hoch(V, V ) is the usual (untruncated) Hochschild complex. Furthermore, in the presence
of an invariant scalar product (which gives rise to a cyclic, or symplectic, A∞ - or L∞ -algebra
V , cf. [7]) we have the following homotopy fiber sequences:
(1.3)
V
f
/ CE(V )
g
/ CE(V ),
in the L∞ -case; here CE(V ) is the usual (untruncated) Chevalley-Eilenberg complex with trivial
coefficients and CE(V ) is its truncated version and
(1.4)
V
f
/ CHoch(V, V )
g
/ CHoch(V, V ),
2000 Mathematics Subject Classification. 18D50, 57T30, 81T18, 16E45.
Key words and phrases. Differential graded Lie algebra, Maurer-Cartan element, A-infinity algebra, graph
homology, Morita equivalence.
The second author is partially supported by an EPSRC research grant.
1
in the A∞ -case; here CHoch(V ) is the usual (untruncated) cyclic Hochschild complex and
CHoch(V ) is its truncated version.
We show that in all four cases the map f can be lifted to an L∞ -map f possessing a certain
universal property which, roughly, says that f takes any Maurer-Cartan element ξ in V into the
Maurer-Cartan element in CE(V, V ), (or Hoch(V, V ), CE(V ), CHoch(V, V )) which corresponds
to the L∞ -structure on V obtained by twisting with ξ. The notions of ∞-Maurer-Cartan
elements and twisted ∞-structures are well-known and appear in many papers on homological
algebra, rational homotopy theory, mirror symmetry and deformation theory, cf. for example
[8, 10, 3, 4]. It seems hard to find a paper collecting all necessary facts about ∞-Maurer-Cartan
elements and twistings and so we give all definitions and proofs of the results that we need.
We do, however, make use of the technology of formal noncommutative (symplectic) geometry
devised by Kontsevich [9] and treated at length in [7] in the context of infinity structures.
The obtained results are used to produce a map from the Chevalley-Eilenberg homology of a
cyclic L∞ - or A∞ -algebra into the corresponding version of a graph complex.
Finally, we show that the sequences (1.1), (1.2), (1.3), (1.4) are homotopy fiber sequences
of L∞ -algebras (as opposed to simply complexes); this makes a connection to a recent work of
Fiorenza and Manetti [2], as well as an earlier work by Voronov on derived brackets [14, 15].
The paper is organized as follows. The remaining part of the introduction reviews some
results and terminology from L∞ - and A∞ -algebras and their cohomology theories. Section 2
introduces the notion of Maurer-Cartan space in the infinity-context and establishes some basic
properties needed later on; these results are undoubtedly known to experts but difficult to find
in the literature. The construction of the L∞ -map f and a relation to the graph homology
are given in Section 3 and the last section establishes a connection to the Fiorenza-Manetti
construction and applications to deformation theory.
1.1. Acknowledgement. The authors wish to thank T. Schedler, for useful comments on a
preliminary version of this paper, and the referee, for many helpful corrections and suggestions.
1.2. Notation and conventions. In this paper we work in the category of Z/2-graded vector
spaces (also known as super-vector spaces) over a field k of characteristic zero. However all our
results (with obvious modifications) continue to hold in the Z-graded context. The adjective
‘differential graded’ will mean ‘differential Z/2-graded’ and will be abbreviated as ‘dg’. A
(commutative) differential graded (Lie) algebra will be abbreviated as (c)dg(l)a. We will often
invoke the notion of a formal (dg) vector space; this is just an inverse limit of finite-dimensional
vector spaces. An example of a formal space is V ∗ , the k-linear dual to a discrete vector space V .
A formal vector space comes equipped with a topology and whenever we deal with a formal
vector space all linear maps from or into it will be assumed to be continuous; thus we will
always have V ∗∗ ∼
= V . All of our unmarked tensors are understood to be taken over k. If V
is a discrete space and W = lim← Wi is a formal space we will write V ⊗ W for lim← V ⊗ Wi ;
thus for two discrete spaces V and U we have Hom(V, U ) ∼
= U ⊗ V ∗ . For a Z/2-graded vector
space V = V0 ⊕ V1 the symbol ΠV will denote the parity reversion of V ; thus (ΠV )0 = V1 while
(ΠV )1 = V0 . The symbol Sn stands for the symmetric group on n letters; it is understood to
act on the nth tensor power of a Z/2-graded vector space V ⊗n by permuting the tensor factors.
1.3. Recollections on L∞ - and A∞ -algebras. Our basic reference on L∞ - and A∞ -algebras
and their cohomology is [7]; we now recall some of the results and terminology from that
paper. A non-unital formal (c)dga is an inverse limit of finite-dimensional nilpotent non-unital
(c)dgas. We will, however, always work with unital cdgas, and a formal cdga will then mean
a cdga obtained by adjoining a unit to a formal non-unital cdga; thus our formal cdgas will be
augmented. The augmentation ideal of an augmented cdga A will be denoted by A+ . There is
an unfortunate clash of terminology: our notion of formality is different to the corresponding
notion in rational homotopy theory. The main examples of formal dgas and cdgas will be
completed tensor and symmetric algebras on (possibly formal) graded vector spaces V ; these
will be denoted by T̂ V and ŜV respectively.
2
Morphisms between and derivations of formal algebras will not be required to be compatible
with augmentations. Note however that the prototypical examples above (and indeed most
of the examples of formal algebras we consider) are local, and hence continuous morphisms
between them automatically preserve augmentations.
An A∞ -algebra structure on a graded vector space V is a continuous odd derivation m of
the formal dga T̂ ΠV ∗ and an L∞ -algebra structure on V is a continuous odd derivation m
of the formal cdga ŜΠV ∗ ; additionally m is required to square to zero and have no constant
term. The components mi : T i ΠV → ΠV or S i ΠV → ΠV of the dual of the restriction of
m to ΠV ∗ are the structure maps of the corresponding A∞ - or L∞ -structure. If (V, m) and
(V ′ , m) are two A∞ - or L∞ -algebras then a morphism between them is a continuous (c)dga map
(T̂ ΠV ′∗ , m′ ) → (T̂ ΠV ∗ , m) or (ŜΠV ′∗ , m′ ) → (ŜΠV ∗ , m), respectively.
The Hochschild complex of an A∞ -algebra (V, m) is the dg vector space
Hoch(V, V ) := (Der(T̂ ΠV ∗ ), [−, m])
consisting of (continuous) derivations of T̂ ΠV ∗ supplied with the differential d(x) = [x, m] for
x ∈ T̂ ΠV ∗ ; this is clearly a dgla. The truncated Hochschild complex Hoch(V, V ) is the sub-dgla
of Hoch(V, V ) consisting of derivations with vanishing constant term.
Similarly the Chevalley-Eilenberg complex CE(V, V ) of an L∞ -algebra (V, m) is the dg vector
space (Der(ŜΠV ∗ ), [, m]) consisting of (continuous) derivations of ŜΠV supplied with the differential d(x) = [x, m] for x ∈ ŜΠV ∗ ; this is clearly a dgla. The truncated Chevalley-Eilenberg
complex CE(V, V ) is the sub-dgla of CE(V, V ) consisting of derivations with vanishing constant
term.
We stress that the algebras Der(T̂ ΠV ∗ ) and Der(ŜΠV ∗ ) of derivations depend only on the dg
vector space V . On the other hand the complexes Hoch(V, V ) and CE(V, V ) depend on (V, m),
the dg vector space V equipped with an A∞ - or L∞ -algebra structure.
We note here that it is more traditional to call the Hochschild complex or the ChevalleyEilenberg complex of (V, m) the dg vector space Π Hoch(V, V ) or Π CE(V, V ); this has the
unpleasant effect that the dgla structures on these spaces become odd which also introduces
additional signs in various formulas, and this is the reason for our convention; a similar remark
also applies to the cyclic complexes below.
For a usual associative algebra the complex Hoch(V, V ) has the familiar form
V → Hom(V, V ) → . . . → Hom(V ⊗n , V ) → . . . ,
whereas the complex Hoch(V, V ) has the same Hochschild differential but starts with the term
Hom(V, V ). Similarly, the complex CE(V, V ) for a Lie algebra V has the form
V → Hom(V, V ) → . . . → Hom(Λn (V ), V ) → . . . ,
whereas the complex CE(V, V ) has the same Chevalley-Eilenberg differential but starts with
the term Hom(V, V ). It is clear that, generally, an element in Hoch(V, V ) is a (possibly infinite)
linear combination of Hochschild cochains, i.e. multilinear maps (ΠV )⊗n → ΠV , and an element
in CE(V, V ) is a (possible infinite) linear combination of Chevalley-Eilenberg cochains, i.e.
graded-symmetric multilinear maps (ΠV )⊗n → ΠV .
In the presence of a graded symmetric, non-degenerate inner product h, i on V we can define cyclic (also called symplectic) L∞ - and A∞ -algebras by the requirement that the expressions hmi (x1 , . . . , xi ), xi+1 i be invariant with respect to cyclic permutations of the elements
x1 , . . . , xi+1 ∈ ΠV . The corresponding complexes are denoted by CHoch(V, V ), CHoch(V, V ),
CE(V ) and CE(V ) respectively.
So, CHoch(V, V ) is the sub dg subspace of Hoch(V, V ) consisting of cyclically invariant
Hochschild cochains, and CHoch(V, V ) is the corresponding sub dg vector space of CHoch(V, V ).
Likewise, CE(V ) is the sub dg subspace of CE(V, V ) consisting of cyclically invariant ChevalleyEilenberg cochains, and CE(V ) is the corresponding sub dg vector space of CE(V, V ).
Note that CE(V ) admits an alternative interpretation as follows. Consider the maximal
ideal ŜΠV+∗ ⊂ ŜΠV ∗ consisting of formal power series with vanishing constant term. Then the
3
derivation m restricts to ŜΠV+∗ and the dg vector space (ŜΠV+∗ , m) is isomorphic to CE(V ); this
is a consequence of the Poincaré lemma in formal symplectic geometry. Note also that CE(V ) is
essentially (disregarding the shift and the lowest degree term) the Chevalley-Eilenberg complex
of (V, m) with trivial coefficients.
We will need to consider infinity-algebra structures over a formal cdga A. We define an
A-linear A∞ -algebra structure on a graded vector space V to be a continuous odd A-linear
derivation m of the formal dga A ⊗ T̂ ΠV ∗ ; in addition m + dA ⊗ id is required to square to
zero and have vanishing constant term. The components mi : A ⊗ T i ΠV → A ⊗ ΠV of the
A-linear dual of the restriction of m to A ⊗ ΠV ∗ are the structure maps of the corresponding
A∞ - or L∞ -structure; they may be viewed as A-multilinear maps on A ⊗ ΠV . Note that a
(k-linear) A∞ -algebra (V, m) is also an A-linear A∞ -algebra (V, mA ), for any formal cdga A,
by extension of scalars: mA := idA ⊗m (equivalently mA
i := idA ⊗mi ). In a similar way we can
introduce the notion of an A-linear L∞ -algebra structure on a graded vector space V . If, in
addition, V possesses a graded symmetric, non-degenerate inner product, A-linear cyclic A∞ and L∞ -algebra structures on V are defined in an obvious way.
2. Maurer-Cartan functor and twisting in L∞ - and A∞ -algebras
In this section we describe the Maurer-Cartan functor for L∞ - and A∞ -algebras in both cyclic
and non-cyclic cases and associated twisted structures.
Definition 2.1. Let V be a graded vector space.
(1) Let m be an L∞ -structure on V and A be a formal cdga with maximal
A+ . Then
P ideal
1 A ⊗i
m
an even element ξ ∈ A+ ⊗ ΠV is Maurer-Cartan if (dA ⊗ id)(ξ) + ∞
i=1 i! i (ξ ) = 0.
(2) Let m be an A∞ -structure on V and A be a formal dga with maximal
A+ . Then
P ideal
A (ξ ⊗i ) = 0.
an even element ξ ∈ A+ ⊗ ΠV is Maurer-Cartan if (dA ⊗ id)(ξ) + ∞
m
i=1 i
The set of Maurer-Cartan elements in A+ ⊗ ΠV will be denoted by MC(V, A). The correspondence (V, A) 7→ MC(V, A) is functorial in A.
Proposition 2.2.
(1) Let (V, m) be an L∞ -algebra and A be a formal cdga. Then functor A 7→ MC(V, A) is
represented by the formal cdga (ŜΠV ∗ , m).
(2) Let (V, m) be an A∞ -algebra and A be a formal dga. Then functor A →
7 MC(V, A) is
represented by the formal cdga (T̂ ΠV ∗ , m).
Proof. For (1) let f : ŜΠV ∗ → A be a map of formal cdgas. Forgetting the differentials, such a
map is equivalent to a continuous linear map ΠV ∗ → A, which is in turn determines an element
ξf ∈ A ⊗ ΠV . The condition that f commutes with differentials could be rewritten as the
commutative diagram:
(2.1)
ΠV ∗
m∗
/ (SΠV )∗
∼
=
/ ŜΠV ∗
f |ΠV ∗
A
f
d
/A
Note that the canonical isomorphism
(S n ΠV )∗ := ((T n ΠV )Sn )∗ = (T̂ n ΠV ∗ )Sn → (T̂ n ΠV ∗ )Sn =: S n ΠV ∗
identifying invariants with coinvariants has the form
1
(T̂ n ΠV ∗ )Sn ∋ x1 ⊗ . . . ⊗ xn 7→ x1 ⊗ . . . ⊗ xn ∈ (T̂ n ΠV ∗ )Sn .
n!
Taking this into account we obtain that the commutativity of the diagram (2.1) is equivalent
to the Maurer-Cartan equation for ξf :
1 A
(dA ⊗ id)(ξf ) + mA
m (ξf ⊗ ξf ) + . . . = 0
1 (ξf ) +
2! 2
4
⊗i → A ⊗ ΠV is the ith higher bracket in the A-linear L -algebra V .
where mA
∞
i : (A ⊗ ΠV )
We omit the proof of (2) which is similar to, but simpler than, that of (1) in that one does
not to have to worry about identification of Sn -invariants with Sn -coinvariants (and thus, no
factorials will appear in the formulas).
Remark 2.3. There is another version of the MC-functor which is occasionally useful. For an
L∞ -algebra (V, m) it associates to a not necessarily formal cdga A the set ξ ∈ ΠV ⊗ A such that
P
∞ 1 A ⊗i
i=1 i! mi (ξ ) = 0 and similarly when (V, m) is an A∞ -algebra. The formal series above does
not have to converge and to justify this definition one has to impose some additional restrictions
on V , e.g. that it possesses only finitely many non-zero higher products as is the case when V is
a dga or a dgla. In this situation (at least when V is finite-dimensional) the functor MC(V, −)
is still represented by a cdga (SΠV ∗ , m) in the L∞ -case and (T ΠV ∗ , m) in the A∞ -case (notice
the absence of the completion). All statements in this section have obvious analogues in this
context.
Remark 2.4. Let (V, m) be an A∞ -algebra. Viewing m as a map ΠV ∗ → T̂ ΠV ∗ and composing
with the projection T̂ ΠV ∗ → ŜΠV ∗ we obtain a map m : ΠV ∗ → ŜΠV ∗ which, obviously,
gives an L∞ -structure on V (this is an infinity version of the familiar commutator Lie algebra
construction). Furthermore, for a formal cgda A the Maurer-Cartan equations in A ⊗ ΠV are
the same regardless of whether (V, m) is considered as an L∞ - or A∞ -algebra; in other words
an element ξ ∈ A ⊗ ΠV is an L∞ -MC-element if and only if it is an A∞ -MC-element.
The Yoneda lemma now gives:
Corollary 2.5.
(1) Let (V, m) and (V ′ , m′ ) be two L∞ -structures and A be a formal cdga. Then any L∞ map V → V ′ determines and is determined by a morphism of functors (in the variable
A): MC(V, A) → MC(V ′ , A).
(2) Let (V, m) and (V ′ , m′ ) be two A∞ -structures and A be a formal dga. Then any A∞ -map
V → V ′ determines and is determined by a morphism of functors (in the variable A):
MC(V, A) → MC(V ′ , A).
Proof. We only have to note that an L∞ -map V → V ′ is by definition a (continuous) map of
formal cdgas (ŜΠV ′∗ , m′ ) → (ŜΠV ∗ , m) and an A∞ -map V → V ′ is a (continuous) map of
formal dgas (T̂ ΠV ′∗ , m′ ) → (T̂ ΠV ∗ , m).
We will now turn to twistings. Let (V, m) be an L∞ -algebra and A be a formal cdga. We
shall explain how a Maurer-Cartan element ξ ∈ MC(V, A) gives rise to a new L∞ -algebra
(V, mξ ) on the same underlying space; the derivation mξ is obtained by applying an appropriate
automorphism of Der(T̂ ΠV ) to m, or equivalently as a constant shift by ξ in the arguments of
m. Both points of view are useful and so we describe both in detail and explain the relationship
between them.
We may regard ξ as a (formal) A-linear derivation of the formal algebra A ⊗ ŜΠV ∗ ; indeed
ξ could be viewed as a linear function on ΠV ∗ ∼
= 1 ⊗ Π(V ∗ ) ⊂ A ⊗ ŜΠV ∗ and we extend it to
the whole of A ⊗ ŜΠV ∗ by the A-linear Leibniz rule.
Consider now eξ := id +ξ + . . . as an automorphism of A ⊗ ŜΠV ∗ ; it is clear that this formal
power series converges. This automorphism acts by conjugations on derivations of A⊗ ŜΠV ∗ ; in
particular on those of square zero; moreover it turns out that it takes one A-linear L∞ -structure
on V to another such structure; the precise statement is given below.
A similar discussion is applicable to A∞ -algebras as well. Namely, let (V, m) be an A∞ algebra, A be a formal dga and ξ ∈ MC(V, A). We shall view ξ as a (formal) A-linear derivation
of the formal algebra A ⊗ T̂ ΠV ∗ ; indeed ξ could be viewed as a linear function on ΠV ∗ ∼
=
1 ⊗ Π(V ∗ ) ⊂ A ⊗ T̂ ΠV ∗ and we extend it to the whole of A ⊗ T̂ ΠV ∗ by the A-linear Leibniz
rule. Then the automorphism eξ acts on the set of A-linear A∞ -structures on V . Here is the
general formulation:
5
Theorem 2.6.
(1) (a) Let (V, m) be an L∞ -algebra, A be a formal cdga and ξ ∈ A+ ⊗ ΠV . Then mξ :=
eξ mA e−ξ , considered as a derivation of A ⊗ ŜΠV ∗ , has no constant term (and thus,
gives another A-linear L∞ -structure on V ) if and only if ξ is Maurer-Cartan. The
process of passing from m to mξ is called twisting by a Maurer-Cartan element ξ.
(b) For x1 , . . . , xn ∈ ΠV the following formula holds:
∞
X
1 A
ξ
(2.2)
mn (x1 , . . . , xn ) =
m (ξ, . . . , ξ, x1 , . . . , xn ).
i! n+i
i=0
(2) (a) Let (V, m) be an A∞ -algebra, A be a formal dga and ξ ∈ A+ ⊗ ΠV . Then mξ :=
eξ mA e−ξ , considered as a derivation of A ⊗ T̂ ΠV ∗ , has no constant term (and thus,
gives another A-linear A∞ -structure on V ) if and only if ξ is Maurer-Cartan. The
process of passing from m to mξ is called twisting by a Maurer-Cartan element ξ.
(b) For x1 , . . . , xn ∈ ΠV the following formula holds:
∞
X
ξ
(2.3)
mn (x1 , . . . , xn ) =
mA
n+i (ξ, . . . , ξ|x1 , . . . , xn ),
i=0
where
X
(ξ,
.
.
.
,
ξ|x
,
.
.
.
,
x
)
:=
mA
mA
1
n
n+i (z1 , . . . , zn+i ),
n+i
the sum running over all n+i
sequences z1 , . . . , zn+i containing x1 , . . . , xn in order,
n
together with i copies of ξ.
Proof. We will restrict ourselves to proving part (1), the proof for (2) is similar but simpler.
Choose a basis
the corresponding linear coordinates xi in ΠV ∗ . Then we can
P i xi in V and
i
write ξ = i ξ xi where ξ ∈ A, and any element in A ⊗ ŜΠV ∗ can be written as a formal
power series f (x1 , . . . , xn , . . . , ) with coefficients in A. The action of eξ on f is then given as
ξ · f = f (x1 + ξ 1 , . . . , xn + ξ n , . . .). P
Furthermore, for a derivation η :=
f i (x1 , . . . , xn , . . . , )∂xi we have
X
(2.4)
eξ ηe−ξ =
f i (x1 + ξ 1 , . . . , xn + ξ n , . . .)∂xi .
i
Let us now take η =
the A-linear derivation of A ⊗ ŜΠV ∗ corresponding to a given L∞ ξ −ξ
structure on V . We see that the derivation
no constant term (and thus, determines
P i e1 ηe has
another L∞ -structure) if and only if i f (ξ , . . . , ξ n , . . .) = 0.
The right hand side of (2.4) is a sum of monomials of the form
mA ,
fi1 ...in (xi1 + ξ i1 )(xi2 + ξ i1 ) . . . (xin + ξ in )∂xi
where fi1 ,...,in ∈ A. Every such monomial represents a linear map ΠV ∗ → A ⊗ ŜΠV ∗ and
the components of the dual map S n ΠV → A ⊗ ΠV correspond to the matrix elements of mξ .
It follows that equation (2.2)
moreover for n = 0 the constant part of the derivation
P holds;
1
mξ corresponds to m0 := ∞
m
(ξ,
. . . , ξ) and so its vanishing is indeed equivalent to the
i=0 i! i
Maurer-Cartan condition as claimed.
Example 2.7. One of the most important special cases of the construction above concerns the
case when V is a dgla or dga (as opposed to a general L∞ - or A∞ -algebra). In that situation
the tensoring with a formal cdga A is unnecessary and we can consider MC(V ) := {ξ ∈ ΠV :
dξ + 12 [ξ, ξ] = 0} (note that this equation is the same in the associative situation since then
1
2
2 [ξ, ξ] = ξ ) In this situation it is easy to check that the corresponding twisted dga or dgla will
have the same product as V and the twisted differential: dξ (x) = dx + [ξ, x] for x ∈ V .
Remark 2.8. Note that the formulas (2.2) and (2.3) do not contain signs; this happens because
our mi ’s are viewed as multi-linear maps on ΠV ; accordingly ξ is treated as an even element
(or constant derivation) and so permuting it past other element does not give rise to signs.
6
We have the following easy result; in order to formulate it we observe that the definition of
MC(V, A) extends in an obvious way to any A-linear L∞ - or A∞ -algebra V . Here and sometimes
later on we will use the shorthand V ξ (or (A ⊗ V )ξ ) to denote the A-linear L∞ - or A∞ -algebra
obtained by twisting by a Maurer-Cartan element ξ.
Proposition 2.9.
(1) Let V be an L∞ -algebra, A be a formal cdga and ξ ∈ MC(V, A). Then for η ∈ MC(V, A)
we have η − ξ ∈ MC(V ξ , A) and there is a one-to-one correspondence MC(V, A) →
MC(V ξ , A) given by η 7→ η − ξ.
(2) Let V be an A∞ -algebra, A be a formal dga and ξ ∈ MC(V, A). Then for η ∈ MC(V, A)
we have η − ξ ∈ MC(V ξ , A) and there is a one-to-one correspondence MC(V, A) →
MC(V ξ , A) given by η 7→ η − ξ.
Proof. We will only give a proof for (1) since it will also work for (2) without any changes.
Denote by m the given L∞ -structure on V . If η ∈ MC(V, A) then
eη−ξ mξ e−η+ξ = eη−ξ eξ me−ξ e−η+ξ = eη me−η .
Therefore eη−ξ mξ e−η+ξ has no constant term and η − ξ ∈ MC(V ξ , A). This argument is clearly
reversible so η − ξ ∈ MC(V ξ , A) if and only if η ∈ MC(V, A).
Remark 2.10. In the next section we will need the analogue of the above result when V is a
dgla and A is a not necessarily formal (see Remark 2.3). The proof will be the same, verbatim.
The twisted infinity structures are compatible with invariant scalar products in the following
sense.
Proposition 2.11.
(1) Let (V, m) be a cyclic L∞ -algebra, A be a formal cdga and ξ ∈ MC(V, A). Then (V, mξ )
is an A-linear cyclic L∞ -algebra.
(2) Let (V, m) be a cyclic A∞ -algebra, A be a formal dga and ξ ∈ MC(V, A). Then (V, mξ )
is an A-linear cyclic A∞ -algebra.
Proof. The proof is the same for both cases; let us therefore consider the L∞ -case. Recall that
ŜΠ(A ⊗ V )∗ is a formal A-supermanifold with a symplectic structure determined by a given
scalar product on V and that m is a symplectic vector field. The adjoint action of the constant
vector field ξ is a translation; therefore it preserves symplectic vector fields. It follows that mξ
is symplectic as required.
Alternatively, one can check directly, using formulas (2.2) that mξ has the desired cyclic
invariance property.
2.1. Maurer-Cartan elements in Hochschild and Chevalley-Eilenberg complexes. Let
V be a graded vector space and A be a formal cdga. Recall that an A-linear L∞ -structure on
V is an odd (continuous) A-linear derivation m of the formal cdga A ⊗ ŜΠV ∗ without constant
term and of square zero. Put another way, m ∈ MC(g, A), where g = Der0 (ŜΠV ∗ ), the
graded Lie algebra of continuous derivations of ŜΠV ∗ without constant term. Let us consider
the corresponding A-linear dgla (A ⊗ g)m obtained by twisting the graded Lie algebra A ⊗ g
by m; recall that the twisted differential will have the form d(x) = [m, x] for x ∈ A ⊗ g.
By definition, the dgla (A ⊗ g)m is isomorphic to the truncated Chevalley-Eilenberg complex
CEA (V, V ) = A ⊗ CE(V, V ) of the A-linear L-infinity algebra (V, m).
Similarly, an A-linear A∞ -algebra structure on V is a Maurer-Cartan element m in the graded
Lie algebra A ⊗ h, where h = Der0 (T̂ ΠV ∗ ) is the graded Lie algebra of continuous derivations
of the formal dga T̂ ΠV ∗ with vanishing constant term; the twisted dgla (A ⊗ h)m is just the
truncated Hochschild complex HochA (V, V ) = A ⊗ Hoch(V, V ) of (V, m).
These observations have cyclic versions: suppose that (V, m) is an A-linear cyclic L∞ -algebra.
That means that the space V ∗ is supplied with a (graded) symplectic structure and that m is a
symplectic derivation. Consider the Lie subalgebra g̃ of g consisting of symplectic derivations;
7
then m ∈ MC(g̃, A) and the twisted dgla (A⊗g̃)m is identified with the truncated cyclic complex
CEA (V ) = A ⊗ CE(V ) of (V, m).
Similarly if (V, m) is a A-linear cyclic A∞ -algebra then the space V ∗ is supplied with a
(graded) symplectic structure and m is a (noncommutative) symplectic derivation. Consider the
Lie subalgebra h̃ of h consisting of symplectic derivations; then m ∈ MC(g̃, A) and the twisted
dgla (A ⊗ g̃)m is identified with the truncated cyclic complex CHochA (V, V ) = A ⊗ CHoch(V, V )
of (V, m).
Then we have the following result which is a direct consequence of Proposition 2.9.
Proposition 2.12. Let A be a cdga:
(1) (a) Let (V, m) be an L∞ -algebra. Then there is a one-to-one correspondence between
the set of A-linear L∞ -structures on V and MC(CE(V, V ), A); namely any A-linear
L∞ -structure m′ determines an MC-element m′ − m ∈ MC(CE(V, V ), A).
(b) Let (V, m) be a cyclic L∞ -algebra. Then there is a one-to-one correspondence between the set of A-linear cyclic L∞ -structures on V and MC(CE(V ), A); namely
any A-linear cyclic L∞ -structure m′ determines an MC-element m′ −m ∈ MC(CE(V ), A).
(2) (a) Let (V, m) be an A∞ -algebra. Then there is a one-to-one correspondence between
the set of A-linear A∞ -structures on V and MC(Hoch(V, V ), A); namely any Alinear A∞ -structure m′ determines an MC-element m′ − m ∈ MC(Hoch(V, V ), A).
(b) Let (V, m) be a cyclic A∞ -algebra. Then there is a one-to-one correspondence
between the set of A-linear cyclic A∞ -structures on V and MC(CHoch(V, V ), A);
namely any A-linear cyclic A∞ -structure m′ determines an MC-element m′ − m ∈
MC(CHoch(V, V ), A).
Remark 2.13. There is an obvious analogue of the above result in the situation when A is a
not necessarily formal cdga, cf. Remark 2.10.
3. Main construction
In this section we describe an L∞ -map f = (f1 , f2 , . . .) from an L∞ - or A∞ -algebra to its
truncated Chevalley-Eilenberg or Hochschild complex as well as the analogues in the cyclic
situation. The first component f1 is the map f appearing in the homotopy fiber sequences
(1.1)-(1.4).
Theorem 3.1.
(1) (a) Let (V, m) be an L∞ -algebra. Then the maps fr : (ΠV )⊗r → ΠCE(V, V ) given by
the formulas
(3.1)
fr (w1 , . . . , wr )(x1 , . . . , xn ) = mr+n (w1 , . . . , wr , x1 , . . . , xn ),
(wi , xi ∈ ΠV ).
define an L∞ -map f : V → CE(V, V ). The map f is characterized by the property
that for any formal algebra A an MC-element ξ ∈ MC(V, A) is mapped to an MCelement in MC(CE(V, V ), A) corresponding to the twisting of A ⊗ V by the element
ξ.
(b) Let (V, m) be a cyclic L∞ -algebra. Then the maps fr : (ΠV )⊗r → ΠCE(V ) are
given by the formulas (3.1) define an L∞ -map f : V → CE(V ). The map f is
characterized by the property that for any formal algebra A an MC-element ξ ∈
MC(V, A) is mapped to an MC-element in MC(CE(V ), A) corresponding to the
twisting of A ⊗ V by the element ξ.
(2) (a) Let (V, m) be an A∞ -algebra. Then the maps fr : (ΠV )⊗r → ΠHoch(V, V ) given by
the formulas
X 1
mr+n (σ(w1 ⊗ . . . ⊗ wr )|x1 , . . . , xn ).
(3.2)
fr (w1 , . . . , wr )(x1 , . . . , xn ) =
r!
σ∈Sr
8
define an L∞ -map f : V → Hoch(V, V ). Here mr+n (wi1 , . .. , wir |x1 , . . . , xn ) :=
P
±mr+n (z1 , . . . , zr+n ) where the sum ranges over all r+n
shuffles of the ses
quences wi1 , . . . , wir and x1 , . . . , xn , and the sign is determined by the Koszul sign
rule. The map f is characterized by the property that for any formal algebra A
an MC-element ξ ∈ MC(V, A) is mapped to an MC-element in MC(Hoch(V, V ), A)
corresponding to the twisting of A ⊗ V by the element ξ.
(b) Let (V, m) be an cyclic A∞ -algebra. Then the maps fr : (ΠV )⊗r → ΠCHoch(V, V )
given by the formulas (3.2) define an L∞ -map f : V → CHoch(V, V ). The map
f is characterized by the property that for any formal algebra A an MC-element
ξ ∈ MC(V, A) is mapped to an MC-element in MC(CHoch(V, V ), A) corresponding
to the twisting of A ⊗ V by the element ξ.
Remark 3.2. The maps f in the L∞ - and A∞ -contexts are compatible in the following sense.
Let (V, m) be an A∞ -algebra; it could be regarded as an L∞ -algebra, cf. Remark 2.4.
Furthermore, extending this observation, one sees that there are maps of dglas Hoch(V, V ) →
CE(V, V ) and Hoch(V, V ) → CE(V, V ). Then straightforward inspection shows that there is a
commutative diagram of L∞ -algebras and maps:
f
V
/ Hoch(V, V )
/ Hoch(V, V )
/ CE(V, V )
/ CE(V, V )
∼
=
V
f
Moreover, a similar statement also holds in the cyclic context.
3.1. Proof of Theorem 3.1: L∞ -algebra case. Let (V, m) be an L∞ -structure on a graded
vector space V . To construct an L∞ -map V → CE(V, V ) it is sufficient to exhibit a map
MC(V, A) → MC(CE(V, V ), A)
for any formal cdga A, which is functorial in A. Indeed, the functor A 7→ MC(V, A) is represented
by the cdga (ŜΠV ∗ , m) and thus, by Yoneda’s lemma such a functorial map is induced by a
cdga map ŜΠCE(V, V )∗ → ŜΠV ∗ i.e. the desired L∞ -map.
Let ξ ∈ MC(V, A); it gives rise via twisting to the A-linear L∞ -algebra V ξ and hence,
by Proposition 2.12 (more precisely, its version described in Remark 2.3), to an MC element
mξ − m ∈ MC(CE(V, V ), A). Note that while in general neither m nor mξ is contained in
A+ ⊗ ΠCE(V, V ), their difference is. So the correspondence ξ 7→ mξ − m defines the desired
map MC(V, A) → MC(CE(V, V ), A).
We will now describe the map f explicitly. To do this we perform the above argument
for
P i
the universal example A = (ŜΠV ∗ , m). Consider the canonical MC-element ξ =
w ⊗ wi ∈
ŜΠV ∗ ⊗ ΠV , where {wi } is a basis of ΠV and {wi } its dual basis in ΠV ∗ ⊂ ŜΠV ∗ . The
corresponding twisted A-linear L∞ -structure on A ⊗ V is given by


∞
X
1 ŜΠV ∗ 
m
ξ, . . . , ξ , x1 , . . . , xn 
mξn (x1 , . . . , xn ) =
| {z }
r! r+n
r=0
r
!
∞
X
X
X
X
1 ŜΠV ∗
=
m
w i1 ⊗ w i1 ,
w i2 ⊗ w i2 , . . . ,
w i r ⊗ w i r , x1 , . . . , x n
r! r+n
r=0
i1
i2
ir
X 1
(3.3)
wir . . . wi1 ⊗ mr+n (wi1 , . . . , wir , x1 , . . . , xn ),
=
r!
r≥0
i1 ,...,ir
for x1 , . . . , xn ∈ ΠV .
9
This structure is interpreted as an MC-element ξ˜ = (ξ˜n ) in the dgla
M
ŜΠV ∗ ⊗ CE(V, V ) =
ŜΠV ∗ ⊗ Hom(S n ΠV, ΠV ),
n≥1
where
ξ˜n (x1 , . . . , xn ) =
X 1
wir . . . wi1 ⊗ mr+n (wi1 , . . . , wir , x1 , . . . , xn ).
r!
r≥1
i1 ,...,ir
(Note that the only the r > 0 terms of the sum (3.3) contribute here.) Finally, interpreting the
MC-element ξ˜ ∈ ŜΠV ∗ ⊗ CE(V, V ) as an L∞ -map f = (fr ) from V into
M
Hom(S n ΠV, ΠV ),
CE(V, V ) =
n≥1
we obtain the desired formula (3.1). The factorial disappears in (3.1) for the same reason that
it appears in the proof of part (1) of Proposition 2.2.
Now assume that V is a cyclic L∞ -algebra. Then the above L∞ -map V → CE(V, V ) factors
through CE(V ). This follows directly from the fact that a Maurer-Cartan twisting of a cyclic
L∞ -algebra is again cyclic, cf. Proposition 2.11.
3.2. Proof of Theorem 3.1: A∞ -algebra case. Now we turn to the closely related associative
case. Let (V, m) be an A∞ -algebra. We will construct an L∞ -map from the associated L∞ algebra (V, m) (cf. Remark 2.4) to the dgla Hoch(V, V ).
To construct such a map it suffices to exhibit a map
MC(V, A) → MC(Hoch(V, V ), A)
for any formal cdga A, which is functorial in A. Indeed, the functor A 7→ MC(V, A) is represented
by the cdga (ŜΠV ∗ , m) and thus, by Yoneda’s lemma, such a functorial map is induced by a
cdga map ŜΠHoch(V, V )∗ → ŜΠV ∗ i.e. the desired L∞ -map.
Let ξ ∈ MC(V, A); it gives rise to a twisted A-linear A∞ -algebra (A⊗V )ξ . Hence it determines
the MC-element mξ − m in the dgla A ⊗ Hoch(V, V ). So the correspondence ξ 7→ mξ − m defines
the desired map MC(V, A) → MC(Hoch(V, V ), A).
We will now describe the map f := (f1 , f2 , . . .) explicitly. Consider the canonical MC-element
P i
ξ=
w ⊗ wi ∈ ŜΠV ∗ ⊗ ΠV , where {wi } is a basis of ΠV and {wi } its dual basis in ΠV ∗ ⊂
ŜΠV ∗ . The corresponding twisted A-linear A∞ -structure on ŜΠV ∗ ⊗ V is given by the product


∞
X
ŜΠV ∗ 
mξn (x1 , . . . , xn ) =
mr+n
ξ, . . . , ξ |x1 , . . . , xn 
| {z }
r=0
r
!
∞
X
X
X
X
∗
ŜΠV
=
mr+n
w i1 ⊗ w i1 ,
wir ⊗ wir |x1 , . . . , xn
w i2 ⊗ w i2 , . . . ,
r=0
(3.4)
=
X
i1
i2
ir
wir . . . wi1 ⊗ mr+n (wi1 , . . . , wir |x1 , . . . , xn ) ,
r≥0
i1 ,...,ir
for x1 , . . . , xn ∈ ΠV . This structure is interpreted as a MC-element ξ˜ = (ξ˜n ) in the dgla
M
ŜΠV ∗ ⊗ Hom(T n ΠV, ΠV ),
ŜΠV ∗ ⊗ Hoch(V, V ) =
n≥1
where
ξ˜n (x1 , . . . , xn ) =
X
wir . . . wi1 ⊗ mr+n (wi1 , . . . , wir |x1 , . . . , xn ).
r≥1
i1 ,...,ir
10
(Note that the only the r > 0 terms of the sum (3.4) contribute here.) Finally, interpreting the
MC-element ξ˜ ∈ ŜΠV ∗ ⊗ Hoch(V, V ) as an L∞ -map f = (fr ) from V into
M
Hom(T n ΠV, ΠV ),
Hoch(V, V ) =
n≥1
we obtain the desired formula (3.2).
Now assume that V is a cyclic A∞ -algebra. Then the above L∞ -map V → Hoch(V, V ) factors
through CHoch(V, V ), since a Maurer-Cartan twisting of a cyclic A∞ -algebra is again cyclic.
3.3. Application to graph homology. We now outline one application of the constructions in this section to graph homology. Let (V, m) be an L∞ -algebra which we assume to
be finite-dimensional for simplicity and consider, as in Section 2.1, the graded Lie algebra
g = Der0 (ŜΠV ∗ ) consisting of (continuous) derivations of ŜΠV ∗ vanishing at zero. Note that g
has the same underlying space as CE(V, V ) but vanishing differential ; particularly it does not
depend on the L∞ -structure on V . Consider the universal twisting of V ; the L∞ -structure on
ŜΠV ∗ ⊗ V corresponding to the twisting by the canonical element. This L∞ -structure can itself
be viewed as an element in MC(g, (ŜΠV ∗ , m)). Note, however, that this element does not lie in
(ŜΠV ∗ )+ ⊗ Πg and thus, we are in the framework of Remark 2.3.
It follows that there is a map of from the CE complex of g to that of V . At this stage we
pass to the dual homological complexes CE∗ (g) and CE∗ (V ); thus CE∗ (V ) ∼
= SΠV with the
differential dual to [−, m] and CE∗ (g) is the standard CE homological complex of the graded
Lie algebra g. The switch to the dual language is not necessary, strictly speaking, but it is more
natural for the interpretation of the result in terms of graph homology.
Note that both CE∗ (V ) and CE∗ (g) are dg coalgebras. It follows that there is a map of dg
coalgebras: χ : CE∗ (V ) → CE∗ (g). Since the coalgebra CE∗ (g) is cogenerated by the component
Πg it follows that such a map is determined by the composition CE∗ (V ) → CE(g) → Πg;
abusing the notation we will denote the latter map also by χ. Furthermore, χ can be viewed as
⊗r
a collection
Q∞ χ = (χn1 . . . , χr . . .) where χr : (ΠV ) → Πg. Moreover, since g could be identified
with n=1 Hom(S V, V ) the element χr (v1 , . . . , vr ) is determined by its value on every collection
x1 , . . . xn ∈ ΠV for n = 0, 1, . . ..
Clearly, the same arguments also work in the A∞ situation, and also in the cyclic versions of
both. We have the following result whose proof is essentially the same as that of Theorem 3.1:
Proposition 3.3.
(1) Let V be a finite-dimensional graded vector space:
(a) Let g = Der0 ŜΠ∗ V be the graded Lie algebra of formal vector fields on V with
vanishing constant term and m ∈ MC(g, k) be an MC-element in g giving V the
structure of an L∞ -algebra. Then there is a map of dg coalgebras χ : CE∗ (V ) →
CE∗ (g) determined by a collection of maps χr : (ΠV )⊗r → Πg, r = 1, 2, . . . such
that
χr (w1 , . . . , wr )(x1 , . . . xn ) = mn+r (w1 , . . . , wr , x1 , . . . , xn ).
where wi , xi ∈ ΠV ; r ≥ 0 and n ≥ 1.
(b) Let h = Der0 T̂ Π∗ V be the graded Lie algebra of formal noncommutative vector
fields on V with vanishing constant term and m ∈ MC(h, k) be an MC-element in
h giving V the structure of an A∞ -algebra. Then there is a map of dg coalgebras
χ : CE∗ (V ) → CE∗ (h) determined by a collection of maps χr : (ΠV )⊗r → Πh, r =
1, 2, . . . such that
X 1
mn+r (σ(w1 ⊗ . . . ⊗ wr )|x1 , . . . , xn ).
χr (w1 , . . . , wr )(x1 , . . . xn ) =
r!
σ∈Sr
where wi , xi ∈ ΠV ; r ≥ 0 and n ≥ 1.
(2) Let V be a finite-dimensional graded vector space with an inner product (and thus, ΠV
is a symplectic vector space):
11
(a) Let g̃ = Der0 ŜΠ∗ V be the graded Lie algebra of formal symplectic vector fields
on V with vanishing constant term and m ∈ MC(g̃, k) be an MC-element in g̃
giving V the structure of a cyclic L∞ -algebra. Then there is a map of dg coalgebras
χ : CE∗ (V ) → CE∗ (g̃) determined by a collection of maps χr : (ΠV )⊗r → Πg̃, r =
1, 2, . . . such that
χr (w1 , . . . , wr )(x1 , . . . xn ) = mn+r (w1 , . . . , wr , x1 , . . . , xn ).
where wi , xi ∈ ΠV ; r ≥ 0 and n ≥ 1.
(b) Let h̃ = Der0 T̂ Π∗ V be the graded Lie algebra of formal noncommutative symplectic
vector fields on V with vanishing constant term and m ∈ MC(h̃, k) be an MCelement in h̃ giving M the structure of an A∞ -algebra. Then there is a map of dg
coalgebras χ : CE∗ (V ) → CE∗ (h̃) determined by a collection of maps χr : (ΠV )⊗r →
Πh̃, r = 1, 2, . . . such that
X 1
mn+r (σ(w1 ⊗ . . . ⊗ wr )|x1 , . . . , xn ).
χr (w1 , . . . , wr )(x1 , . . . xn ) =
r!
σ∈Sr
where wi , xi ∈ ΠV ; r ≥ 0 and n ≥ 1.
Remark 3.4. The statement of Proposition 3.3 appears almost tautologically identical to that
of Theorem 3.1; however the essential difference between these results is that we allow the value
r = 0 in the formulas for χ (but not in the formulas for f (3.1), (3.2) ). This has the effect that
χ does not give an L∞ -map from V into g. In fact, χ could be thought of as a curved L∞ -map,
cf. the last section concerning curved L∞ -algebras. However, having such a map is sufficient to
get an induced map on the CE homology.
The above result is particularly interesting when V is a cyclic L∞ - or A∞ -algebra. In that case
letting the dimension of V go to infinity we obtain the stable Lie algebra g∞ whose homology
CE(g∞ ) is essentially Kontsevich’s graph complex in the L∞ -case and the complex of ribbon
graphs in the A∞ -case, [9, 5, 6]. Therefore, we obtain a map from CE∗ (V ) to the appropriate
version of the graph complex. This map is related to, but quite different from another construction, the so-called direct construction of Kontsevich which associates a graph homology class
to V itself. The precise connection between these constructions is the subject of a forthcoming
paper.
4. Homotopy fiber sequences of L∞ -algebras
In the previous section we constructed an L∞ -map f : V → CE(V, V ) for an L∞ -algebra V
which fits into a homotopy fiber sequence of dg vector spaces
V
f
/ CE(V, V )
/ CE(V, V ),
and similarly in the A∞ -algebra case. It is natural to ask whether these are actually homotopy
fiber sequences in the category of L∞ -algebras. In this section we show that this is indeed the
case and explain the relation to work of Voronov and of Fiorenza-Manetti.
We begin by recalling the relevant notions. Note first that for any L∞ -algebra (V, m) there is
defined another such V [z, dz] := V ⊗ k[z, dz] where k[z, dz] is the polynomial de Rham algebra
on the interval; the evaluation maps |z=a : k[z, dz] → k, a = 0, 1, induce the corresponding
evaluation maps |z=a : V [z, dz] → V .
Definition 4.1. Let V, W be two L∞ -algebras and f, g : V → W be two maps between them.
A homotopy between f and g is an L∞ -algebra map s : V → W [z, dz] such that s|z=0 = f and
s|z=1 = g.
12
Note that a map V → W is by definition a (formal) cdga map ŜΠW ∗ → ŜΠV ∗ ; the latter is
equivalent to an element in MC(W, ŜΠV ∗ ). We shall be particularly interested in the situation
when W is a dgla in which case (ŜΠV ∗ )+ ⊗ W is a (formal) dgla. Then there is the following
characterization of homotopy:
Proposition 4.2. Let V be an L∞ -algebra, W be a dgla and f, g ∈ MC(W, ŜΠV ∗ ) be two
corresponding L∞ -maps. Then f and g are homotopic if and only if there exists an element
ξ ∈ (ŜΠV ∗ )+ ⊗ ΠW such that f = eξ ge−ξ .
Proof. This follows from a well-known characterization of homotopy in formal (or pronilpotent)
dglas, cf. for example [1], Theorem 4.4.
Furthermore we could form the homotopy category of dglas by formally inverting quasiisomorphisms. There is also the notion of a homotopy fiber sequence.
Definition 4.3. For a dgla map f : g → h the homotopy fiber of f is defined as Hf := {(g, h) ∈
g ⊕ h[z, dz] : h|z=0 = 0, h|z=1 = f (g)}; the projection onto g is a dgla map Hf → g. The
sequence Hf → g → h as well as any 3-term sequence of dg Lie algebras homotopy equivalent
to it will be called a homotopy fiber sequence of dglas.
f
/ h the composite map Hf → h is homotopic
/g
For a homotopy fiber sequence Hf
to zero; moreover it possesses a semi-universal property: any map φ : a → h of dglas which is
homotopic to zero after composing with f factors through Hf . Namely, if s : a → h[z, dz] is a
nullhomotopy of f ◦ φ, then the factorization is given by (φ, s) : a → Hf ⊂ g ⊕ h[z, dz]. We will
need a slight extension of this property: suppose that a is an L∞ -algebra and φ is an L∞ -map
from a into g whose composition with f is nullhomotopic. Then the same formula as above
gives an L∞ -map from a into Hf .
The dgla homotopy fiber Hf should be compared with the dg homotopy fiber Cf = g ⊕ Πh
with differential d defined as follows. For (g, h) ∈ g ⊕ Πh, set d(g, h) = (∂g, −∂h + f (g)).
The following result is standard (see, e.g., [2, Lemma 3.2]).
Proposition 4.4. The dg vector space Cf is a retract of Hf , via the maps i : Cf → Hf and
π : Hf → Cf , given by
i(g, h) = (g, f (g)z + hdz);
Z 1
h(z, dz)).
π(g, h(z, dz)) = (g,
0
Remark 4.5. Note that dg fiber Cf does not itself support a dgla structure. However Fiorenza
and Manetti [2] use the retraction in Proposition 4.4 to transport the dgla structure of Hf to
an L∞ -algebra structure on Cf .
Note that the category of dglas is a subcategory of the category of L∞ -algebras, and this
inclusion induces an equivalence of homotopy categories obtained by inverting respectively strict
or L∞ -quasi-isomorphisms. We can now define a homotopy fiber sequence of L∞ -algebras.
Definition 4.6. The sequence of L∞ -algebras and maps
a→g→h
is called homotopy fiber if it is L∞ -quasi-isomorphic to a homotopy fiber sequence of dglas.
It is clear that the ‘homotopy fiber’ of an L∞ -map f : g → h (i.e. an L∞ -algebra a which
f
/ h ) is determined uniquely up to a non/g
can be included in a fiber sequence a
canonical isomorphism in the homotopy category and that it possesses the analogous semiuniversal property as the homotopy fiber of a dgla map.
Theorem 4.7.
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(1) (a) Let V be an L∞ -algebra; then the sequence
f
V
g
/ CE(V, V )
/ CE(V, V ))
is a fiber sequence of L∞ -algebras.
(b) Let V be a cyclic L∞ -algebra; then the sequence
f
V
g
/ CE(V, V )
/ CE(V, V ))
is a fiber sequence of L∞ -algebras.
(2) (a) Let V be an A∞ -algebra; then the sequence
f
V
/ Hoch(V, V )
g
/ Hoch(V, V )
is a fiber sequence of L∞ -algebras.
(b) Let V be a cyclic A∞ -algebra; then the sequence
V
f
/ CE(V, V )
g
/ CE(V, V )
is a fiber sequence of L∞ -algebras.
Proof. We restrict ourselves with proving the statement about L∞ -algebras; the other parts
are proved similarly. Denote by α = g ◦ f : V → CE(V, V ) the composite L∞ -map. Let
us first prove that α is homotopic to zero. To this end we view α as an MC element in the
dgla ŜΠV ∗ ⊗ Π CE(V, V ). The zero map is P
represented by the CE differential 1 ⊗ [, m] in this
dgla. Observe that α = eξ de−ξ where ξ =
wi ⊗ wi where wi and wi are dual bases in ΠV
and ΠV ∗ respectively, and wi is viewed as a (constant) derivation of ŜΠV ∗ . It follows from
Proposition 4.2 that α is indeed homotopic to zero. The homotopy has the following explicit
form: αz = ezξ de−zξ + ξdz.
The arguments similar to those used to derive formulas (3.1) translate this homotopy to an
L∞ -map s = {sr } from V to CE∗ (V, V )[z, dz]:
(
z r fr (w1 , . . . , wr ), if r > 1
sr (w1 , . . . , wr ) =
zf1 (w) + ŵdz, if r = 1
Here ŵ refers to the constant derivation of ŜΠV ∗ associated to w viewed as an element in
Π CE(V, V ).
∗
The L∞ -map s gives rise to an L∞ -map s̃ = {s̃r } from V to Hg ⊂ CE (V, V ) ⊕ Π CE∗ (V, V ).
Here s̃r : (ΠV )⊗n → ΠHg :
s̃r (w1 , . . . , wr ) = (fr (w1 , . . . , wr ), sr (w1 , . . . , wr )).
It remains to show that s̃1 : ΠV → ΠHg is a quasi-isomorphism. For this, it suffices to prove
that the composite map π ◦ s̃1 : ΠV → ΠCg is a quasi-isomorphism. We have
Z 1
s1 (w)) = (f1 (w), ŵ).
(π ◦ s̃1 )(w) = (f1 (w),
0
Observe that CE(V, V ) is the dg mapping cone of f1 : V → CE(V, V ), and g is the canonical
map of CE(V, V ) into that cone. So we conclude the argument by appealing to the following
general fact.
Lemma 4.8. Let
X
f
g
/Y
/ cone(f )
be a standard cofiber sequence of dg vector spaces and let Cg be the homotopy fiber of g; recall
that Cg has Y ⊕ ΠY ⊕ X as its underlying graded vector space. Then the map h : X → Cg
defined by h(x) = (f (x), 0, x) is a quasi-isomorphism.
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We now describe the relationship of Theorem 4.7 to Voronov’s work on higher derived brackets
[14, 15]. We are indebted to the referee for bringing the precise connection to our attention.
Voronov’s starting point is an inclusion K ֒→ L of dgla’s with abelian complement W . He
shows that L ⊕ ΠW may be endowed with an L∞ -algebra structure so that the projection map
L ⊕ ΠW is a replacement of the original inclusion K ֒→ L by a fibration. This implies that
ΠW , equipped with Voronov’s higher derived brackets, is a homotopy fiber of K ֒→ L.
The inclusion CE(V, V ) ֒→ CE(V, V ) (and its variations) we have considered in this section
is a particular case of Voronov’s setup, with complementary space W = ΠV . His theory,
applied in this case, shows that the L∞ -algebra (V, m) is a homotopy fiber of the inclusion.
The new statement in Theorem 4.7 is the assertion that one obtains a homotopy fiber sequence
V → CE(V, V ) → CE(V, V ), in which the first map is the map explicitly described in formulas
(3.1) and (3.2).
Our explicit realisation of a homotopy fiber sequence may be compared with the main result
of Fiorenza and Manetti [2]. Given an arbitrary dgla map K → L, they construct a homotopy
fiber sequence C → K → L of L∞ -algebras, where the underlying dg vector space of the
homotopy fiber L∞ -algebra C is the homotopy fiber of K → L in the category of dg vector
spaces.
4.1. Curved L∞ -algebras and deformations. We will now outline one application of the
above theorem which, roughly, could be formulated as saying that an L∞ -algebra V controls the
deformations of V as an L∞ -algebra which are trivial as deformations of curved L∞ -algebras. A
similar statement can, of course, be made in the A∞ or cyclic contexts; we will restrict ourselves
to treating the L∞ case only.
Definition 4.9. A curved L∞ -algebra structure on a graded vector space V is a (continuous)
odd derivation m of the algebra ŜΠV ∗ such that [m, m] = 0.
Note that such a derivation can be written as m = m0 + m1 + . . . where mi is an odd map
Ŝ(ΠV ∗ ) → ΠV ∗ which is equivalent to a collection of Sn -invariant maps (ΠV )⊗n → ΠV subject
to some constraints coming from the identity [m, m] = 0. The difference with the notion of
an L∞ -algebra is that m is not required to have zero constant term; consequently m0 could be
non-zero and m1 does not necessarily square to zero.
For a curved L∞ -algebra (V, m) we could form its Chevalley-Eilenberg complex CE(V, V );
its underlying space is Der Ŝ(V ) and the differential is defined as the commutator with m, just
as in the uncurved case. The condition [m, m] = 0 ensures that the differential squares to
zero, however there is no analogue of CE(V, V ) since [, m] does not restrict to the subspace of
derivations with the vanishing constant term. Clearly, CE(V, V ) is a dgla. This dgla controls
deformations of V ; let us briefly recall the general setup of (derived) deformation theory, cf. for
example [11].
Let g be a dgla (or, more generally, an L∞ -algebra). Let Def g be the functor associating to a
formal cdga A the pointed set M C (g, A) of MC-elements in A+ ⊗ g modulo gauge equivalence.
Here we say that two elements x0 , x1 ∈ MC(g, A) are gauge equivalent if they are homotopic,
i.e. there exists an MC element X ∈ MC(g[z, dz], A) such that X|z=0 = x0 and X|z=1 =
x1 . The base point in Def g is the gauge class of the zero MC element. The functor Def g
is homotopy invariant in g, which means that a quasi-isomorphism g → g′ induces a natural
bijection Def g (A) ∼
= Def g′ (A).
Definition 4.10. A deformation of an L∞ -algebra V over a formal cdga A is an MC-element in
A+ ⊗ CE(V, V ). Two such deformations are called equivalent if the corresponding MC-elements
are gauge equivalent. Similarly a deformation of an curved L∞ -algebra V over a formal cdga
A is an MC-element in A+ ⊗ CE(V, V ). Two such deformations are called equivalent if the
corresponding MC-elements are gauge equivalent.
Now let g → h → a be a homotopy fiber sequence of L∞ -algebras. Associated to it is the
sequence of functors Def g → Def h → Def a which is fiber in the sense that the preimage in Def h
15
of the base point in Def a lies in the image of Def g . This could be easily seen by using the
homotopy invariance of Def and reducing to a short exact sequence of dglas. Note that the
map Def g → Def h is not necessarily injective; in fact it has an action of the group of homotopy
self-equivalences of the functor Def a , however explaining this would take us too far afield and
we refrain from giving further details.
We obtain the following result, which is an immediate consequence of theorem 4.7.
Corollary 4.11. A deformation of an L∞ -algebra V over a formal cdga A which is trivial as
its deformation as a curved L∞ -algebra is always given by a twisting with some MC-element
ξ ∈ MC(V, A).
In conclusion note that the constructions of this paper have an analogue in conventional
homotopy theory. Let X be a pointed space; denote by Map(X, X) and Map(X, X) the space
of all (respectively pointed) self-maps of X. There is an obvious fiber sequence of spaces
Map(X, X) → Map(X, X) → X
which gives rise to the homotopy fiber sequence
Ω(X) → Map(X, X) → Map(X, X)
where Ω(X) is the space of based loops on X. This is parallel to any of the fiber sequences
(1.1), (1.2), (1.3) or (1.4). Note that there does not seem to be a simple description of the map
Ω(X) → Map(X, X).
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Centre for Mathematical Science, City University London, London EC1V 0HB, UK
E-mail address: [email protected]
University of Leicester, Department of Mathematics, Leicester LE1 7RH, UK.
E-mail address: [email protected]
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